A Straight line parallel to the line $\mathrm{y}=\sqrt{3}\mathrm{x}$ passes through Q(2,3) cuts the line 2x + 4y - 27 = 0 at P, Find the length of PQ.

$\text{ Slope of the line }\mathrm{y}=\sqrt{3}\mathrm{x}\text{ is }\sqrt{3}$
$\text{ Slope of }\overleftrightarrow{\mathrm{PQ}}\text{ is }\sqrt{3}$
$\text{ inclination of }\overleftrightarrow{\mathrm{PQ}}\text{ is }{60}^{\circ }$
If PQ = r then the coordiantes of P are
$\begin{array}{l}\left({\mathrm{x}}_1+\mathrm{rcos}\mathrm{\theta },{\mathrm{y}}_1+\mathrm{rsin}\mathrm{\theta }\right)=\left(2+\mathrm{rcos}{60}^0,3+\mathrm{rsin}{60}^0\right)\\ =\left(2+\frac{\mathrm{r}}{2},3+\frac{\sqrt{3}\mathrm{r}}{2}\right)\end{array}$
$\mathrm{Since} \mathrm{P}\text{ lies on the line }2\mathrm{x}+4\mathrm{y}-27=0$ then
$\begin{array}{r}2\left(2+\frac{\mathrm{r}}{2}\right)+4\left(3+\frac{\sqrt{3}\mathrm{r}}{2}\right)-27=0\\ \mathrm{r}=2\sqrt{3}-1\Rightarrow \mathrm{PQ}=2\sqrt{3}-1\\ \\ \end{array}$